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Alan Baker's theorem (1966–68) giving explicit lower bounds for linear forms in logarithms of algebraic numbers, the basis of effective transcendence and Diophantine methods. Claims about the theorem itself, its refinements and effective applications (e.g., Catalan/Pillai-type results), and comparisons of other problems' difficulty to it belong here; general Diophantine approximation without conne

Proving the Collatz conjecture is at least as difficult as proving Baker's theorem on linear forms in logarithms.

SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Collatz conjecture Number theoryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The lower bound on |2^n − 3^m| implied by the weak Collatz conjecture is far weaker than the bound Baker's theorem gives.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Gaps between powers of 2 and 3 Number theory Collatz conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

All known proofs that the Collatz map has no m-cycles for small m rely on Baker's theorem on linear forms in logarithms.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Collatz conjecture Number theory Gaps between powers of 2 and 3importance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

Every known nontrivial lower bound for |2^n − 3^m| relies on transcendence theory such as Baker's method.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Number theoryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

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